Some remarks concerning Voevodsky s nilpotence conjecture

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1 Some remarks concerning Voevodsky s nilpotence conjecture The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Bernardara, Marcello et al. Some Remarks Concerning Voevodsky s Nilpotence Conjecture. Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2018, 738 (May 2018): De Gruyter Walter de Gruyter GmbH Version Final published version Accessed Mon Jan 14 07:22:41 EST 2019 Citable Link Terms of Use Detailed Terms Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

2 J. reine angew. Math. 738 (2018), Journal für die reine und angewandte Mathematik DOI /crelle De Gruyter 2018 Some remarks concerning Voevodsky s nilpotence conjecture By Marcello Bernardara at Toulouse, Matilde Marcolli at Pasadena and Gonçalo Tabuada at Cambridge, MA Abstract. In this article we extend Voevodsky s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky s original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds. 1. Introduction and statement of results Let k be a base field and F a field of coefficients of characteristic zero. Voevodsky s nilpotence conjecture. In a foundational work [36], Voevodsky introduced the smash-nilpotence equivalence relation nil on algebraic cycles and conjectured its agreement with the classical numerical equivalence relation num. Concretely, given a smooth projective k-scheme X, he stated the following: Conjecture V.X/. We have Z.X/ F = nil D Z.X/ F = num. Thanks to the work of Kahn Sebastian, Matsusaka, Voevodsky, and Voisin (see, for instance, [16, 30, 36, 37] and also [1, Section ]), the above conjecture holds in the case of curves, surfaces, and abelian 3-folds (when k is of characteristic zero). G. Tabuada was partially supported by the National Science Foundation CAREER Award # and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações). M. Marcolli was partially supported by the NSF grants DMS and PHY Sections 3 5 are a revised version of half of an old preprint [26] by M. Marcolli and G. Tabuada. The remaining work is a recent collaboration between M. Bernardara and G. Tabuada.

3 300 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture Noncommutative nilpotence conjecture. A dg category A is a category enriched over dg k-vector spaces; see Section 2.1. Following Kontsevich [18 20], A is called smooth if it is perfect as a bimodule over itself and proper if for any two objects x; y 2 A we have P i dim H i A.x; y/ < 1. The classical example is the unique dg enhancement perf dg.x/ of the category of perfect complexes perf.x/ of a smooth projective k-scheme X; see Lunts and Orlov [25]. As explained in Section , the Grothendieck group K 0.A/ of every smooth proper dg category A comes endowed with a -nilpotence equivalence relation nil and with a numerical equivalence relation num. Motivated by the above conjecture, we state the following: Conjecture V NC.A/. We have K 0.A/ F = nil D K 0.A/ F = num. Our first main result is the following reformulation of Voevodsky s conjecture: Theorem 1.1. Conjecture V.X/ is equivalent to Conjecture V NC.perf dg.x//. Theorem 1.1 shows us that when restricted to the commutative world, the noncommutative nilpotence conjecture reduces to Voevodsky s original conjecture. Making use of this noncommutative viewpoint, we now address Voevodsky s nilpotence conjecture in several cases. Quadric fibrations. Let S be a smooth projective k-scheme and q W Q! S a flat quadric fibration of relative dimension n with Q smooth. Recall from Kuznetsov [23] (see also [5]) the construction of the sheaf C 0 of even parts of the Clifford algebra associated to q. Recall also from [23] that when the discriminant divisor of q is smooth and n is even (resp. odd), we have a discriminant double cover es! S (resp. a square root stack bs) equipped with an Azumaya algebra B 0. Our second main result allows us to decompose Conjecture V.Q/ into simpler pieces: Theorem 1.2. (i) We have The following hold: V.Q/ V NC.perf dg.s; C 0 // C V.S/: (ii) When the discriminant divisor of q is smooth and n is even, we have V.Q/ V.eS/ C V.S/: As a consequence, V.Q/ holds when dim.s/ 2, and becomes equivalent to V.eS/ when S is an abelian 3-fold and k is of characteristic zero. (iii) When the discriminant divisor of q is smooth and n is odd, we have V.Q/ V NC.perf dg.bs; B 0 // C V.S/: As a consequence, Conjecture V.Q/ becomes equivalent to V NC.perf dg.bs/; B 0 / when dim.s/ 2. This latter conjecture holds when dim.s/ 2. Remark 1.3. The (rational) Chow motive of a quadric fibration q W Q! S was computed by Vial in [35, Theorem 4.2 and Corollary 4.4]. In the particular case where dim.s/ 2,

4 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture 301 it consists of a direct sum of submotives of smooth projective k-schemes of dimension at most two. This motivic decomposition provides an alternative geometric proof of Conjecture V.Q/. We will rely on this argument to prove the last statement of item (iii). In the particular case where S is a curve and k is algebraically closed, we provide also a categorical proof of this last statement; see Remark 6.5. The fact that V NC.perf dg.bs/; B 0 / holds when dim.s/ 2 will play a key role in the proof of Theorem 1.4 below. Intersection of quadrics. Let X be a smooth complete intersection of r quadric hypersurfaces in P m. The linear span of these r quadrics gives rise to a hypersurface Q P r 1 P m, and the projection into the first factor is a flat quadric fibration q W Q! P r 1 of relative dimension m 1. Theorem 1.4. (i) We have The following holds: V.X/ V NC.perf dg.p r 1 ; C 0 //: (ii) When the discriminant divisor of q is smooth and m is odd, we have As a consequence, V.X/ holds when r 3. V.X/ V.A P r 1 /: (iii) When the discriminant divisor of q is smooth and m is even, we have V.X/ V NC.perf dg.1 P r 1 ; B 0 //: This latter conjecture holds when r 3 and k is algebraically closed. Remark 1.5. The (rational) Chow motive of a complete intersection X was computed in [8, Theorem 2.1] in the particular cases where r 2 or r D 3 and m is odd. It consists of a direct sum of submotives of smooth projective k-schemes of dimension at most one. This motivic decomposition provides an alternative proof of Conjecture V.X/. A similar argument holds in the case where r D 3 and m is even. Remark 1.6 (Relative version). Theorem 1.4 has a relative analogue with X replaced by a generic relative complete intersection X! S of r quadric fibrations Q i! S of relative dimension m 1; consult [5, Definition 1.2.4] for details. Items (i), (ii), and (iii), hold similarly with P r 1 replaced by a P r 1 -bundle T! S, with V.A P r 1 / replaced by V.eT / C V.S/, and with V NC.perf dg.1 P r 1 ; B 0 // replaced by V NC.perf dg.bt ; B 0 // C V.S/, respectively. Note that thanks to the relative item (ii), Conjecture V.X/ holds when r D 2 and S is a curve. Linear sections of Grassmannians. two classes of schemes: Following Kuznetsov [21], consider the following (i) Let X L be a generic linear section of codimension r of the Grassmannian Gr.2; W / (with W D k 6 ) under the Plücker embedding, and Y L the corresponding dual linear section of the cubic Pfaffian Pf.4; W / in P.ƒ 2 W /. For example when r D 3, X L is a Fano 5-fold; when r D 4, X L is a Fano 4-fold; and when r D 6, X L is a K3 surface of degree 14 and Y L a Pfaffian cubic 4-fold. Moreover, X L and Y L are smooth whenever r 6.

5 302 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture (ii) Let X L be a generic linear section of codimension r of the Grassmannian Gr.2; W / (with W D k 7 ) under the Plücker embedding, and Y L the corresponding dual linear section of the cubic Pfaffian Pf.4; W / in P.ƒ 2 W /. For example when r D 5, X L is a Fano 5-fold; when r D 4, X L is a Fano 4-fold; when r D 8, Y L is a Fano 4-fold; and when r D 9, Y L is a Fano 5-fold. Moreover, X L and Y L are smooth whenever r 10. Theorem 1.7. Let X L and Y L be as in the above classes (i) (ii). Under the assumption that X L and Y L are smooth, we have V.X L / V.Y L /: This conjecture holds when r 6 (class (i)), and when r 6 and 8 r 10 (class (ii)). Remark 1.8. To the best of the authors knowledge, Theorem 1.7 proves Voevodsky s nilpotence conjecture in new cases. Linear sections of determinantal varieties. Let U and V be k-vector spaces of dimensions m and n, respectively, with n m, and 0 < r < m an integer. Following [7], consider the universal determinantal variety Zm;n r P.U V / given by the locus of matrices M W U! V of rank at most r. Its Springer resolution is denoted by X r m;n WD P.U V /! Gr.r; U /; where U stands for the tautological subbundle on Gr.r; U /. Under these notations, we have the following class of schemes: (i) Let X L be a generic linear section of codimension c of Xm;n r under the map X r m;n! P.U V /; and Y L the corresponding dual linear section of X m X m r m;n r m;n! P.U V /: under the map Remark 1.9. As explained in [7, Section 3], X L and Y L are smooth crepant categorical resolution of singularities of.zm;n r / L and.zm;n m r/ L, respectively. For example when m D n D 4 and r D 1, X L is a.6 c/-dimensional section of P 3 P 3 under the Segre embedding, and Y L the resolution of the dual.c 2/-dimensional determinantal quartic. In the same vein, when m D n D 4 and r D 2, X L is an.11 c/-dimensional section of the self-dual orbit of 4 4 matrices of rank 2, and Y L the.c 5/-dimensional dual section. Moreover, the following holds: (a) When c 7, X L is a Fano.11 c/-fold and dim.y L / 2. (b) When c D 8, X L and Y L are dual Calabi Yau 3-folds. (c) When c 9, Y L is a Fano (c 5)-fold and dim.x L / 2. For further example, consult [7, Section 3.3] and well as [7, Tables 1 and 2].

6 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture 303 Theorem When X L and Y L are as in the above class (i), we have V.X L / V.Y L /: This conjecture holds when dim.x L / 2 or dim.y L / 2. Remark To the best of the authors knowledge, Theorem 1.10 proves Voevodsky s nilpotence conjecture in new cases. Homological projective duality. Making use of Kuznetsov s theory of homological projective duality (HPD) [22], Theorem 1.7 admits the following generalization: let X be a smooth projective k-scheme equipped with an ample line bundle O X.1/. Note that O X.1/ gives rise to a morphism X! P.V /, where V WD H 0.X; O X.1//. Let Y be the HP-dual of X, O Y.1/ the associated ample line bundle, and Y! P.V / the associated morphism. Assume that perf.x/ admits a Lefschetz decomposition perf.x/ D ha 0 ; : : : ; A n.n/i with respect to O X.1/, i.e. a semi-orthogonal decomposition of perf.x/ such that A 0 : : : A n and A i.i/ WD A i O.i/. Assume also that Conjecture V NC.A dg 0 / holds, where Adg 0 stands for the dg enhancement of A 0 induced from perf dg.x/; see Section 2.2. Finally, let L V be a subspace such that the linear sections X L WD X P.V / P.L/ and Y L WD Y P.V / P.L? / are of expected dimension dim.x L / D dim.x/ dim.l/ and dim.y L / D dim.y / dim.l? /, respectively. Theorem smooth, we have Let X L and Y L be as above. Under the assumption that X L and Y L are V.X L / V.Y L /: Remark Theorem 1.12 reduces to Theorem 1.7 (resp. to Theorem 1.10) in the particular case of Grassmannian Pfaffian (resp. determinantal) homological projective duality; consult [21] (resp. [7]) for details. Moishezon manifolds. A Moishezon manifold X is a compact complex manifold such that the field of meromorphic functions on each component of X has transcendence degree equal to the dimension of the component. As proved by Moishezon [31], X is a smooth projective C-scheme if and only if it admits a Kähler metric. In the remaining cases, Artin [3] showed that X is a proper algebraic space over C. Let Y! P 2 be one of the non-rational conic bundles described by Artin and Mumford in [4], and X! Y a small resolution. In this case, X is a smooth (not necessarily projective) Moishezon manifold. Theorem Conjecture V NC.perf dg.x// holds for the above resolutions. Remark The proofs of Theorems 1.2, 1.4, 1.7, 1.10, 1.12, and 1.14 are based on the study of a smooth projective k-scheme (or algebraic space) X via semi-orthogonal decompositions of its category of perfect complexes perf.x/; see Bondal Orlov [9], Huybrechts [14], and Kuznetsov [24] for instance. This approach allows the reduction of Voevodsky s Conjecture V.X/ to several noncommutative conjectures V NC one for each piece of the semi-orthogonal decomposition. We believe this provides a new tool for the proof of Voevodsky s conjecture as well as of its generalization to algebraic spaces.

7 304 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture 2. Preliminaries 2.1. Dg categories. A differential graded category (in short, dg category) A is a category enriched over dg k-vector spaces; consult Keller [17] for details. For example, every (dg) k-algebra A gives naturally rise to a dg category A with a single object. Let dgcat be the category of small dg categories. Recall from [17, Section 3] the construction of the derived category D.A/ of A. This triangulated category admits arbitrary direct sums and we will write D c.a/ for the full subcategory of compact objects. A dg functor A! B is called a Morita equivalence if it induces an equivalence D.A/! D.B/. Finally, let us write A B for the tensor product of dg categories Perfect complexes. Given a stack X and a sheaf of O X -algebras G, let Mod.X; G / be the Grothendieck category of sheaves of (right) G -modules, D.X; G / WD D.Mod.X; G // the derived category of G, and perf.x; G / the subcategory of perfect complexes. As explained in [17, Section 4.4], the derived category D dg.ex/ of an abelian (or exact) category Ex is defined as the (Drinfeld s) dg quotient C dg.ex/=ac dg.ex/ of the dg category of complexes over Ex by its full dg subcategory of acyclic complexes. Hence, let us write D dg.x; G / for the dg category D dg.ex/ with Ex WD Mod.X; G / and perf dg.x; G / for the full dg subcategory of perfect complexes. Lemma 2.1. Let X be a smooth projective k-scheme, and let perf.x/ D ht 1 ; : : : ; T n i be a semi-orthogonal decomposition. In this case, the dg categories Ti dg (where Ti dg stands for the dg enhancement of T i induced from perf dg.x/) are smooth and proper. Proof. Let Ho.dgcat/ be the localization of dgcat with respect to the class of Morita equivalences. The tensor product of dg categories gives rise to a symmetric monoidal structure on dgcat which descends to Ho.dgcat/. Moreover, as proved in [10, Theorem 5.8], the smooth and proper dg categories can be characterized as those objects of Ho.dgcat/ which are dualizable. Note that the canonical inclusion T dg i,! perf dg.x/ and projection perf dg.x/! T dg i dg functors express Ti dg as a direct factor of perf dg.x/ in Ho.dgcat/. Hence, since perf dg.x/ is smooth and proper, we conclude that Ti dg is also smooth and proper nilpotence equivalence relation. Let A be a dg category. An element ŒM of the Grothendieck group K 0.A/ WD K 0.D c.a// is called -nilpotent if there exists an integer n > 0 such that ŒM n D 0 in K 0.A n /. This gives rise to a well-defined equivalence relation nil on K 0.A/ and on its F -linearization K 0.A/ F Numerical equivalence relation. Let A be a smooth proper dg category. As explained in [27, Section 4], the pairing.m; N / 7! P i. 1/i dim Hom Dc.A/.M; N Œi/ gives rise to a well-defined bilinear form. ; / on K 0.A/. Moreover, the left and right kernels of. ; / are the same. An element ŒM of the Grothendieck group K 0.A/ is said to be numerically trivial if.œm ; ŒN / D 0 for all elements ŒN 2 K 0.A/. This gives rise to an equivalence relation num on K 0.A/ and consequently on K 0.A/ F. When A D perf dg.x/, with X a smooth projective k-scheme, and F D Q, this equivalence relation reduces, via the Chern character K 0.X/ Q! CH.X/ Q, to the classical numerical equivalence relation on the Chow ring CH.X/ Q.

8 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture Motives. We assume the reader is familiar with the categories of Chow motives Chow.k/ F and numerical motives Num.k/ F ; see [1, Chapitre 4]. The Tate motive will be denoted F.1/. In the same vein, we assume some familiarity with the categories of noncommutative Chow motives NChow.k/ F and noncommutative numerical motives NNum.k/ F ; consult the surveys [28, Sections 2 3] and [32, Section 4], and the references therein. Recall from [1, Chapitre 4] that NNum.k/ F is the idempotent completion of the quotient of NChow.k/ F by its largest -ideal 1), and that Hom NChow.k/F.k; A/ ' K 0.A/ F. 3. Orbit categories and -nilpotence Let C be an F -linear additive rigid symmetric monoidal category. Orbit categories. Given a -invertible object O 2 C, recall from [33, Section 7] the construction of the orbit category C= O. It has the same objects as C and morphisms Hom C= O.a; b/ WD M Hom C.a; b O j /: j 2Z The composition law is induced from C. By construction, C= O is F -linear, additive, and comes equipped with a canonical projection functor W C! C= O. Moreover, is endowed with a natural 2-isomorphism ı. O/ ) and is 2-universal among all such functors. As proved in [33, Lemma 7.3], C= O inherits from C a symmetric monoidal structure making symmetric monoidal. On objects it is the same. On morphisms it is defined as the unique bilinear pairing M Hom C.a c;.b d/ O j / j 2Z Hom C.a; b O j / M j 2Z Hom C.c; d O j /! M j 2Z which sends the pair to.a f r! b O r ; c g s! d O s /.f g/.rcs/ W a c f r g s! b O r d O s '.b d/ O.rCs/ : -nilpotence. The nil -ideal of C is defined as nil.a; b/ WD ¹f 2 Hom C.a; b/ W f n D 0 for n 0º: By construction, nil is a -ideal. Moreover, all its ideals nil.a; a/ Hom C.a; a/ are nilpotent; see [2, Lemma (ii)]. As a consequence, the -functor C! C= nil is not only F -linear and additive but moreover conservative. Furthermore, since idempotents can be lifted along nilpotent ideals (see [6, Section III, Propposition 2.10]), C= nil is idempotent complete whenever C is idempotent complete. 1) Different from the entire category NChow.k/ F.

9 306 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture Compatibility. Let C be a category and O 2 C a -invertible objects as above. Proposition 3.1. There exists a canonical F -linear additive -equivalence making the following diagram commute: (3.2) C= nil C C= O.C= nil /= O '.C= O /= nil. Proof. The existence of the F -linear additive -functor follows from the fact that (3.3) C! C= O!.C= O /= nil vanishes on the nil -ideal and also from the natural 2-isomorphism between (3.3) ı. O/ and (3.3). Note that the functor is the identity on objects and sends ¹Œf j º j 2Z to Œ¹f j º j 2Z. Clearly, it is full. The faithfulness is left as an exercise. 4. -nilpotence of motives By construction, the categories Chow.k/ F and NChow.k/ F are F -linear, additive, rigid symmetric monoidal, and idempotent complete. Let us denote by and Voev.k/ F WD Chow.k/ F = nil NVoev.k/ F WD NChow.k/ F = nil the associated quotients. They fit in the following sequences: and Chow.k/ F! Voev.k/ F! Num.k/ F NChow.k/ F! NVoev.k/ F! NNum.k/ F : The relation between all these motivic categories is the following: Proposition 4.1. There exist F -linear additive fully-faithful -functors R; R nil ; R N making the following diagram commute: (4.2) Chow.k/ F Chow.k/ F = F.1/ R NChow.k/ F Voev.k/ F Voev.k/ F = F.1/ R nil NVoev.k/ F Num.k/ F Num.k/ F = F.1/ R N NNum.k/ F.

10 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture 307 Proof. The outer commutative square, with R; R N both F -linear additive fully-faithful -functors, was built in [29, Theorem 1.13]. Consider now the zoomed diagram (4.3) Chow.k/ F = F.1/ R Chow.k/ F = F.1/ NChow.k/ F Voev.k/ F = F.1/ ' R= nil.chow.k/ F = F.1/ /= nil NVoev.k/ F Num.k/ F = F.1/ Num.k/ F = F.1/ R N NNum.k/ F. By definition, R nil WD R= nil ı. Since R is an F -linear additive fully-faithful -functor, we conclude that R nil is also an F -linear additive fully-faithful -functor. The commutativity of the bottom squares of diagram (4.3) follows from the fact that Num.k/ F = F.1/ identifies with the quotient of Chow.k/ F = F.1/ by its largest -ideal N ; consult [29, Proposition 3.2] for details. 5. Proof of Theorem 1.1 Note first that we have the following natural isomorphisms: Hom Voev.k/F = F.1/.Spec.k/; X/ ' Z.X/ F = nil ; Hom Num.k/F = F.1/.Spec.k/; X/ ' Z.X/ F = num : As a consequence, Conjecture V.X/ becomes equivalent to the injectivity of (5.1) Hom Voev.k/F = F.1/.Spec.k/; X/ Hom Num.k/F = F.1/.Spec.k/; X/: Given a smooth and proper dg category A, we have also natural isomorphisms Hom NVoev.k/F.k; A/ ' K 0.A/ F = nil ; Hom NNum.k/F.k; A/ ' K 0.A/ F = num : Hence, Conjecture V NC.A/ becomes equivalent to the injectivity of (5.2) Hom NVoev.k/F.k; A/ Hom NNum.k/F.k; A/: Now, recall from [33, Theorem 1.1] that the image of X under the composed functor R ı identifies naturally with the noncommutative Chow motive perf dg.x/. Similarly, the image of Spec.k/ under R ı identifies with perf dg.spec.k// which is Morita equivalent to k. As a consequence, since the functors R nil and R N are fully-faithful, the bottom right-hand side square of diagram (4.2) gives rise to the following commutative diagram: Hom Voev.k/F = F.1/.Spec.k/; X/ ' Hom NVoev.k/F.k; perf dg.x// (5.1) Hom Num.k/F = F.1/.Spec.k/; X/ ' (5.2) Hom NNum.k/F.k; perf dg.x//. Using the above reformulations of Conjectures V and V NC, we conclude finally that Conjecture V.X/ is equivalent to Conjecture V NC.perf dg.x//.

11 308 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture Item (i). decomposition 6. Proof of Theorem 1.2 As proved by Kuznetsov in [23, Theorem 4.2], one has the semi-orthogonal perf.q/ D hperf.s; C 0 /; perf.s/ 1 ; : : : ; perf.s/ n i with perf.s/ i WD q perf.s/ O Q=S.i/. Note that we have perf.s/ i ' perf.s/ for every i. Using [8, Proposition 3.1], one then obtains a direct sum decomposition in NChow.k/ F (6.1) perf dg.q/ ' perf dg.s; C 0 / perf dg.s/ perf dg.s/; ƒ n copies where perf dg.s; C 0 / stands for the dg enhancement of perf.s; C 0 / induced from perf dg.q/. Note that by Lemma 2.1, the dg category perf dg.s; C 0 / is smooth and proper. Since the inclusion of categories perf.s; C 0 /,! perf.q/ is of Fourier Mukai type ([23, Proposition 4.9]), its kernel K 2 perf.s Q; C op 0 O X/ gives rise to a Fourier Mukai Morita equivalence ˆKdg W perf dg.s; C 0/! perf dg.s; C 0 /: Hence, we can replace in the above decomposition (6.1) the dg category perf dg.s; C 0 / by the canonical one perf dg.s; C 0 / (see Section 2.2). Finally, using the above description (5.2) of the noncommutative nilpotence conjecture, one concludes that (6.2) V NC.perf dg.q// V NC.perf dg.s; C 0 // C V NC.perf dg.s//: The proof follows now automatically from Theorem 1.1. Item (ii). As proved by Kuznetsov in [23, Proposition 3.13], perf.s; C 0 / is Fourier Mukai equivalent to perf.es; B 0 /. Hence, the above equivalence (6.2) reduces to (6.3) V NC.perf dg.q// V NC.perf dg.es; B 0 // C V NC.perf dg.s//: Since B 0 is a sheaf of Azumaya algebras and F is of characteristic zero, the canonical dg functor perf dg.es/! perf dg.es; B 0 / becomes an isomorphism in NChow.k/ F ; see [34, Theorem 2.1]. Consequently, Conjecture V NC.perf dg.es; B 0 // reduces to Conjecture V NC.perf dg.es//. The proof follows now from Theorem 1.1. Item (iii). As proved by Kuznetsov in [23, Proposition 3.15], perf.s; C 0 / is Fourier Mukai equivalent to perf.bs; B 0 /. Hence, the above equivalence (6.2) reduces to (6.4) V NC.perf dg.q// V NC.perf dg.bs; B 0 // C V NC.perf dg.s//: The proof of the first claim follows now from Theorem 1.1. Let us now prove the second claim, which via (6.4) is equivalent to the proof of Conjecture V NC.perf dg.q//. Thanks to Vial [35, Theorem 4.2 and Corollary 4.4], the rational Chow motive M Q.Q/ of Q decomposes as M Q.Q/ D M.S/.n dim.s// Q N, where N stands for a submotive of a smooth projective k-scheme of dimension dim.s/. Therefore, when dim.s/ 2, Conjecture V.Q/ D V NC.perf dg.q// holds. Remark 6.5. Assume that S is a smooth projective curve and that k is algebraically closed. In this remark we provide a categorical proof of the second claim of item (iii) of Theorem 1.2. Thanks to the work of Graber Harris Starr [13], the fibration q W Q! S admits

12 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture 309 a section. Making use of it, we can perform reduction by hyperbolic splitting in order to obtain a conic bundle q 0 W Q 0! S; consult [5, Section 1.3] for details. The sheaf C0 0 of even parts of the associated Clifford algebra is such that the categories perf.s; C 0 / and perf.s; C0 0 / are Fourier Mukai equivalent; see [5, Remark 1.8.9]. As a consequence, using (6.2) and the fact that dim.s/ D 1, we obtain the following equivalence: (6.6) V NC.perf dg.q// V NC.perf dg.q 0 //: Since S is a curve, Q 0 is a surface. Therefore, conjecture (6.6) holds. 7. Proof of Theorem 1.4 Item (i). As proved by Kuznetsov in [23, Theorem 5.5], we have a Fourier Mukai equivalence perf.x/ ' perf.p r 1 ; C 0 / when m 2r C 1 D 0, the semi-orthogonal decomposition perf.x/ D hperf.p r 1 ; C 0 /; O X.1/; : : : ; O X.m 2r C 1/i when m 2r C 1 > 0, and a dual semi-orthogonal decomposition of perf.p r 1 ; C 0 / (containing a copy of perf.x/ and exceptional objects) when m 2r C 1 < 0. The proof of the case m 2r C 1 D 0 is clear. Let us now prove the case m 2r C 1 > 0; the proof of the case m 2r C 1 < 0 is similar. Using [8, Proposition 3.11], one obtains the following direct sum decomposition in NChow.k/ F : perf dg.x/ ' perf dg.p r 1 ; C 0 / perf dg.k/ perf dg.k/ : ƒ.m 2rC1/ copies Thanks to Lemma 2.1, the dg category perf dg.p r 1 ; C 0 / is smooth and proper. 2) Since the inclusion perf.p r 1 ; C 0 /,! perf.x/ is of Fourier Mukai type (see [23, Proposition 4.9]), an argument similar to the one of the proof of Theorem 1.2 (i) shows us that V NC.perf dg.x// V NC.perf dg.p r 1 ; C 0 // C V NC.perf dg.k//: The proof follows now automatically from Theorem 1.1. Items (ii) (iii). The proofs are similar to those of items (ii) (iii) of Theorem Proof of Theorems 1.7 and 1.10 Assume that X L and Y L are as in classes of (i) (ii) of Theorem 1.7 (resp. as in class (i) of Theorem 1.10). As proved by Kuznetsov in [21, Sections 10 11] (resp. in [7, Theorem 3.4]), one of the following three situations occurs: (a) there is a semi-orthogonal decomposition perf.x L / D hperf.y L /; E 1 ; : : : ; E n i, where E i are exceptional bundles on X L, (b) there is a semi-orthogonal decomposition perf.y L / D hperf.x L /; E 0 1 ; : : : ; E0 n i, where E0 i are exceptional bundles on Y L, (c) there is a Fourier Mukai equivalence between perf.x L / and perf.y L /. 2) In the case m 2r C 1 < 0, these properties follow from the existence of a fully faithful Fourier Mukai functor perf.p r 1 ; C 0 /! perf.q/, with Q P r 1 P m a smooth hypersurface.

13 310 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture Therefore, the equivalence of V.X L / and V.Y L / is clear in situation (c). Since the inclusions of categories perf.y L /,! perf.x L / (situation (a)) and perf.x L /,! perf.y L / (situation (b)) are of Fourier Mukai type, a proof similar to the one of Theorem 1.2 (i) shows us that the equivalence of V.X L / and V.Y L / also holds in situations (a) (b). Note that this concludes the proof of Theorem 1.10 since Conjecture V.X L / (resp. V.Y L /) holds when dim.x L / 2 (resp. dim.y L / 2). Let us now focus on class (i) of Theorem 1.7. The smooth projective k-schemes X L and Y L are of dimensions 8 r and r 2, respectively. Hence, V.Y L / holds when r 4 and V.X L / when r D 6. When r D 5, X L (and Y L ) is a Fano 3-fold. As explained by Gorchinskiy and Guletskii in [12, Section 5], the Chow motive of X admits a decomposition into Lefschetz motives and submotives of curves. This implies that V.X L / also holds. Let us now focus on class (ii) of Theorem 1.7. The smooth projective k-schemes X L and Y L are of dimensions 10 r and r 4, respectively. Hence, V.Y L / holds when r 6 and V.X L / when r 8. This achieves the proof. 9. Proof of Theorem 1.12 Following Kuznetsov [22, Section 4], let us denote by a i the orthogonal complement of A ic1 in A i ; these are called the primitive subcategories in [22, Section 4]. Since Conjecture V NC.A dg 0 / holds, we hence conclude by induction that Conjectures V NC.A dg i / and V NC.a dg i / also hold for every i. Thanks to HPD (see [22, Theorem 6.3]), perf.y / admits a Lefschetz decomposition perf.y / D hb m. m/; : : : ; B 0 i with respect to O Y.1/ such that the primitive subcategories b i coincide (via a Fourier Mukai functor) with the primitive subcategories a i. Consequently, V NC.b dg i / holds for every i. An inductive argument, starting with b m D B m, allows us then to conclude that Conjecture V NC.B dg i / also holds for any i. Now, thanks once also to HPD (see [22, Thm. 5.3]), there exists also a triangulated category C L and semi-orthogonal decompositions perf.x L / D hc L ; A dim.l/.1/; : : : ; A n.n dim.l//i; perf.y L / D hb m.dim.l? / m/; : : : ; B dim.l? /. 1/; C L i: Moreover, the composed functor perf.x L /! C L! perf.y L / is of Fourier Mukai type. As a conclusion, since X L and Y L are smooth, we can apply Theorem 1.1 and obtain the following chain of equivalences: V.X L / V NC.perf dg.x L // V NC.C dg L / V NC.perf dg.y L // V.Y L /: This achieves the proof. 10. Proof of Theorem 1.14 Thanks to the work of Cossec [11], the conic bundle Y! P 2 has a natural structure of quartic double solid Y! P 3 ramified along a quartic symmetroid D. Via the natural involution on the resolution of singularities of D, one hence obtains an Enriques surface S; consult [11, Section 3] for details. As proved by Zube in [38, Section 5], one has moreover a semiorthogonal decomposition perf.s/ D ht S ; E 1 ; : : : ; E 10 i, where E i are exceptional objects. Let

14 Bernardara, Marcolli and Tabuada, Remarks concerning Voevodsky s nilpotence conjecture 311 us denote by T dg S the dg enhancement of T S induced from perf dg.s/. Thanks to Lemma 2.1, T dg S is smooth and proper. Hence, since S is a surface, an argument similar to the one of the proof of Theorem 1.2 (i) shows us that Conjecture V NC.T dg S / holds. Now, recall from Ingalls and Kuznetsov [15, Section 5.5] the construction of the Fourier Mukai functor ˆ W perf.s/! perf.x/ whose restriction to T S is fully-faithful. As proved in [15, Proposition 3.8 and Theorem 4.3], one has a semi-orthogonal decomposition perf.x/ D hˆ.t S /; E 0 1 ; E0 2 i; where E 0 i are exceptional objects. As a consequence, we obtain the equivalence (10.1) V NC.perf dg.x// V NC.ˆ.T S / dg /; where ˆ.T S / dg stands for the dg enhancement of ˆ.T S / induced from perf dg.x/. Since the kernel K of the above Fourier Mukai functor ˆ gives rise to a Morita equivalence ˆKdg W T dg S! ˆ.T S/ dg ; we conclude that conjecture (10.1) also holds. This achieves the proof. Acknowledgement. The authors are grateful to Bruno Kahn and Claire Voisin for useful comments and answers, as well as to the anonymous referee for useful remarks. References [1] Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panor. Synthèses 17, Société Mathématique de France, Paris [2] Y. André and B. Kahn, Nilpotence, radicaux et structures monoïdales, Rend. Semin. Mat. Univ. Padova 108 (2002), [3] M. Artin, Algebraization of formal moduli. II: Existence of modification, Ann. of Math. (2) 91 (1970), [4] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. (3) 25 (1972), [5] A. Auel, M. Bernardara and M. Bolognesi, Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems, J. Math. Pures Appl. (9) 102 (2014), [6] H. Bass, Algebraic K-theory, W. A. Benjamin, New York [7] M. Bernardara, M. Bolognesi and D. Faenzi, Homological projective duality for determinantal varieties, preprint 2014, [8] M. Bernardara and G. Tabuada, Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives, preprint 2013, [9] A. Bondal and D. Orlov, Derived categories of coherent sheaves, in: International congress of mathematicians. Vol. II: Invited lectures (Beijing 2002), Higher Education Press, Beijing (2002), [10] D.-C. Cisinski and G. Tabuada, Symmetric monoidal structure on noncommutative motives, J. K-Theory 9 (2012), no. 2, [11] F. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, [12] S. Gorchinskiy and V. Guletskii, Motives and representability of algebraic cycles on threefolds over a field, J. Algebraic Geom. 21 (2012), no. 2, [13] T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, [14] D. Huybrechts, Fourier Mukai transforms in algebraic geometry, Oxford Math. Monogr., Clarendon Press, Oxford, [15] C. Ingalls and A. Kuznetsov, On nodal Enriques surfaces and quartic double solids, preprint 2010, arxiv.org/abs/ [16] B. Kahn and R. Sebastian, Smash-nilpotent cycles on abelian 3-folds, Math. Res. Lett. 16 (2009), no. 6,

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